Problem: Evaluate the definite integral. $\int^{-1}_{-5}\left(\dfrac{x^6-25x^4}{x^6}\right)\,dx = $
First, use the power rule: $\begin{aligned}\int^{-1}_{-5}\left(\dfrac{x^6-25x^4}{x^6}\right)\,dx~&=~\int^{-1}_{-5}\left(\dfrac{x^6}{x^6}-\dfrac{25x^4}{x^6}\right)\,dx \\&=~\int^{-1}_{-5}\left(1-25x^{-2}\right)\,dx \\&=(x+25x^{-1})\Bigg|^{-1}_{{-5}}\end{aligned}$ Second, plug in the limits of integration: $[{-1}+25\cdot({-1})^{-1}]-[{-5}+25\cdot({-5})^{-1}] = -26+10 = -16$. The answer: $\int^{-1}_{-5}\left(\dfrac{x^6-25x^4}{x^6}\right)\,dx= -16$